Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,7)(6,8)(10,14)(12,15)(17,18)(19,23)(20,24)(21,22);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,16,12)(17,19)(18,20)(21,24)(22,23);(2,5)(6,8)(10,14)(11,16)(18,22)(19,24);(17,18)(19,20)(21,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,20)(21,24)(22,23);(1,3)(2,6)(4,7)(5,8)(9,12,13,15)(10,11,14,16)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^9|b^2=gh,f^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72233511, #72233515, #72233593, … |
| maximal normal subgroup of | #72233511, #72233515, #72233593, … |
| maximal quotient of | #72261001, #72261002, #72297764, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7616076);